Kernelization of Constraint Satisfaction Problems:A Study through Universal Algebra

A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the Boolean satisfiability problem (SAT), and the constraint satisfaction problem (CSP), there exist many results concerning upper and lower bounds for kernelizability of specific problems, but it is safe to say that we lack general methods to determine whether a given SAT problem admits a kernel of a particular size. This could be contrasted to the currently flourishing research program of determining the classical complexity of finite-domain CSP problems, where almost all non-trivial tractable classes have been identified with the help of algebraic properties. In this paper, we take an algebraic approach to the problem of characterizing the kernelization limits of NP-hard SAT and CSP problems, parameterized by the number of variables. Our main focus is on problems admitting linear kernels, as has, somewhat surprisingly, previously been shown to exist. We show that a CSP problem has a kernel with O(n) constraints if it can be embedded (via a domain extension) into a CSP problem which is preserved by a Maltsev operation. We also study extensions of this towards SAT and CSP problems with kernels with O(n^c) constraints, c>1, based on embeddings into CSP problems preserved by a k-edge operation, k > c. These results follow via a variant of the celebrated few subpowers algorithm. In the complementary direction, we give indication that the Maltsev condition might be a complete characterization of SAT problems with linear kernels, by showing that an algebraic condition that is shared by all problems with a Maltsev embedding is also necessary for the existence of a linear kernel unless NP is included in co-NP/poly

Designing FPT algorithms for cut problems using randomized contractions

We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway Cut-Uncut problems. More precisely, we show the following: • We prove that the parameterized version of the Unique Label Cover problem, which is the base of the Unique Games Conjecture, can be solved in 2O(k 2 log |Σ|)n4 log n deterministic time (even in the stronger, vertex-deletion variant) where k is the number of unsatisfied edges and |Σ | is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of Unique Games where the number of edges allowed not to be satisfied is upper bounded by O( log n) to optimality, which improves over the trivial O(1) upper bound

Algorithmes exponentiels pour l'étiquetage, la domination et l'ordonnancement

This manuscript of Habilitation à Diriger des Recherches enlights some results obtained since my PhD, I defended in 2007. The presented results have been mainly published in international conferences and journals. Exponential-time algorithms are given to solve various decision, optimization and enumeration problems. First, we are interested in solving the L(2,1)-labeling problem for which several algorithms are described (based on branching, divide-and-conquer and dynamic programming). Some combinatorial bounds are also established to analyze those algorithms. Then we solve domination-like problems. We develop algorithms to solve a generalization of the dominating set problem and we give algorithms to enumerate minimal dominating sets in some graph classes. As a consequence, the analysis of these algorithms implies combinatorial bounds. Finally, we extend our field of applications of moderately exponential-time algorithms to scheduling problems. By using dynamic programming paradigm and by extending the sort-and-search approach, we are able to solve a family of scheduling problems.Ce manuscrit d’Habilitation à Diriger des Recherches met en lumière quelques résultats obtenus depuis ma thèse de doctorat soutenue en 2007. Ces résultats ont été, pour l’essentiel, publiés dans des conférences et des journaux internationaux. Des algorithmes exponentiels sont donnés pour résoudre des problèmes de décision, d’optimisation et d’énumération. On s’intéresse tout d’abord au problème d’étiquetage L(2,1) d’un graphe, pour lequel différents algorithmes sont décrits (basés sur du branchement, le paradigme diviser-pour-régner, ou la programmation dynamique). Des bornes combinatoires, nécessaires à l’analyse de ces algorithmes, sont également établies. Dans un second temps, nous résolvons des problèmes autour de la domination. Nous développons des algorithmes pour résoudre une généralisation de la domination et nous donnons des algorithmes pour énumérer les ensembles dominants minimaux dans des classes de graphes. L’analyse de ces algorithmes implique des bornes combinatoires. Finalement, nous étendons notre champ d’applications de l’algorithmique modérément exponentielle à des problèmes d’ordonnancement. Par le développement d’approches de type programmation dynamique et la généralisation de la méthode trier-et-chercher, nous proposons la résolution de toute une famille de problèmes d’ordonnancement